12. Techniques of Integration

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [cos(θ) / (sin²(θ) + sin(θ) − 6)] dθ

Use the method of partial fractions to evaluate the integral.

${\int }_{}^{}\frac{5x^3-x^2+7x-2}{(x^2+1)(x^2+2)}dx$

76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.

76. ∫ [cosθ / (sin³θ - 4sinθ)] dθ

In Exercises 33–38, perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral.

∫ 2y⁴ / (y³ - y² + y - 1) dy

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [1 / √(e^s + 1)] ds

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ 9 dv / (81 − v⁴)

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [(2x³ + x² − 21x + 24) / (x² + 2x − 8)] dx

87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.

A: dx = 2/(1 + u²) du

B: sin x = 2u/(1 + u²)

C: cos x = (1 - u²)/(1 + u²)

88. Evaluate ∫ dx/(2 + cos x).

23-64. Integration Evaluate the following integrals.

54. ∫ (z + 1)/[z(z² + 4)] dz

85. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. More than one integration method can be used to evaluate ∫ (1 / (1 - x²)) dx.

23-64. Integration Evaluate the following integrals.

41. ∫₋₁¹ x/(x + 3)² dx

23-64. Integration Evaluate the following integrals.

57. ∫ (x³ + 5x)/(x² + 3)² dx

Evaluate the integrals in Exercises 39–54.

∫ 1 / (x√x + 9) dx

Evaluate the integrals in Exercises 39–54.

∫ (e⁴t + 2e²t - e^t) / (e²t + 1) dt

23-64. Integration Evaluate the following integrals.

32. ∫ (4x - 2)/(x³ - x) dx